Identification and Estimation of Intensive Margin Effects by Difference-in-Difference Methods

1 Introduction

A decomposition of a binary treatment into extensive and intensive margin effects is of special interest when studying outcomes with a corner solution at zero.^[1] Outcomes with corner solutions include working hours, health expenditures, and trade volumes. The average effect of a treatment on an outcome with a corner solution at zero can be decomposed into 1) the average change in the outcome of those with a positive outcome irrespective of treatment, plus 2) the average outcome of those with a positive outcome in case of treatment and a zero outcome in case of no treatment, minus 3) the average outcome of those with a zero outcome in case of treatment and a positive outcome in case of no treatment [1, 2, 3]. Part 1) represents the weighted causal intensive margin effect. The sum of 2) and 3) captures the weighted causal extensive margin effect.^[2]

Take as an example the effect of the introduction of a partial retirement policy on labor supply. The aim of such a policy is to increase labor market participation of older people and to preserve human capital. Suppose that in the status quo, individuals must withdraw the full pension at a given age, but are allowed to continue working.^[3] Under the partial retirement policy, individuals have the choice between a partial and a full pension, and are allowed to continue working. Claiming a partial pension can be attractive because it increases subsequent pension benefits as a result of an extended contribution period as well as reduced and delayed pension claims. The total effect of such a policy on labor supply might be zero, suggesting that the policy has been ineffective. The zero result, however, could be explained by a positive extensive margin effect that was offset by a negative intensive margin effect. Older workers who would have retired in the absence of a partial retirement policy, may now decide to stay in the labor market. Likewise, individuals who would have worked full-time in the absence of a partial retirement policy, may now decide to work part-time. In such cases the total effect masks interesting subeffects at the extensive and intensive margin.

Even if treatment is randomly assigned, estimating intensive margin effects is challenging. A mean comparison of treatment and control groups with positive outcomes does not identify the causal intensive margin effect without additional assumptions [4]. In the labor supply example, the sample of individuals with positive working hours consists of two groups: 1) the group of individuals with positive working hours irrespective of whether they are treated or not (always-participants), i.e. irrespective of whether they have the possibility to withdraw a partial pension; and 2) the group of individuals with positive working hours only because they are treated (switchers), i.e. only because they have the possibility to withdraw a partial pension, who would not work if they could only withdraw the full pension.^[4] For the causal intensive margin effect, we are only interested in the group of always-participants. Group membership is however not observed in the data, because we observe either the outcome in case of treatment or the outcome in case of no treatment. The unobserved characteristics of always-participants and switchers are likely to be different. Always-participants might be more motivated than switchers. Therefore, average working hours of always-participants are likely higher than average working hours of switchers. As a result, a difference in the means of treated and untreated – conditional on positive working hours – could be the result of differences in these unobserved characteristics, and not because of a causal effect of treatment.

This constitutes a selection problem. In a general setting without random treatment, we are thus faced with two selection problems. The first selection problem is the standard selection problem in observational studies. In the presence of confounding variables, a mean comparison of treated and untreated individuals does not identify the causal effect. The second selection problem arises because we condition on positive outcomes. Difference-in-difference methods were developed to deal with the first selection problem. Using data from pre- and post-treatment periods, difference-in-difference allows for some selection on unobservables. This comes at the cost of making an assumption about outcome trends over time. It seems reasonable to extend the difference-in-difference methodology to include the second selection problem as well.

In this paper, we introduce difference-in-difference methods to estimate the causal intensive margin effect. Compared to standard difference-in-difference estimators, we condition the sample on individuals with positive outcomes.^[5] We derive sufficient conditions under which the causal intensive margin effect is identified. In contrast to standard difference-in-difference methods, two monotonicity assumptions are additionally required to identify the causal intensive margin effect. We apply the difference-in-difference methodology to estimate the causal intensive margin effect of reaching the full retirement age on working hours. Knowledge of the causal intensive margin effect is important to obtain a better understanding of retirement choices, for example to improve the design of future pension reforms. Moreover, we discuss how the identifying assumptions can be motivated in practice.

The main contribution of this paper is to extend the literature on identification and estimation of intensive margin effects by borrowing well established difference-in-difference methods from the policy evaluation literature. The intensive margin effect is of interest in cases where the total effect masks relevant subeffects, e.g. when the extensive and intensive margin effect have different signs. Compared to models for outcomes with corner solutions or selection models, the difference-in-difference estimator on positive outcomes is based on different assumptions; most importantly on a common trend assumption.

This paper is related to the literature on models for outcomes with corner solutions, e.g. Tobit [5, 6] or two-part models [7, 8], and selection models [9]. Moreover, the paper is connected to the literature employing principal stratification following [10] to study causal extensive and intensive margin treatment effects for variables with nonnegative outcomes [2, 1, 3]. This literature decomposes the average treatment effect into a population-weighted sum of treatment effects on always-participants and switchers. Studying outcomes with a corner solution at zero, [2] derives nonparametric bounds for the treatment effects on always-participants and switchers. He further discusses point identification of causal intensive and extensive margin effects in censored regression, selection, and two-part models. [1, 3] analyzes total, extensive, and intensive margin effects in general sample selection models, with the corner solution outcome as a special case. [1] analyzes nonparametric methods to estimate extensive and intensive margin effects, whereas [3] discusses point identification of intensive and extensive margin effects in semiparametric linear models. The notion of principal stratification is also used in instrumental variable approaches [11], and in mediation analysis [12]. In instrumental variable approaches, the stratification is based on the treatment variable (always-takers, compliers, defiers, never-takers), whereas in mediation analysis, the stratification in based on the mediator. In our context, the stratification is based on the outcome variable. More generally, the paper often draws upon [13], who provides a survey on difference-in-difference methods from a potential outcomes perspective.

The remainder of the paper is organized as follows. Section 2 introduces the notation and describes the conventional as well as the causal decomposition of a treatment effect. Identification of the causal intensive margin effect is described in Section 3. Section 4 discusses estimation and inference. An empirical application is presented in Section 5. The last section concludes.

2 Notation and Decomposition of a Treatment Effect

3 Identification

We are interested in the intensive margin average treatment effect on the treated (IMATT),

We will first derive sufficient conditions under which γ_t(x), i.e. the conditional-on-X version of the intensive margin average treatment effect on the treated, is identified. In a second step, we state sufficient conditions under which the conditional-on-X version can be aggregated to E(Yi,t1-Yi,t0|Yi,t1>0,Yi,t0>0,Di=1) .

4 Estimation and Inference

Difference-in-difference estimation requires estimating different conditional expectations. Here we adopt a split sample approach. Let Δ Y_i,t = Y_i,t − Y_{i,t − 1}. For the difference-in-difference on positive outcomes estimator, we first estimate

using ordinary least squares. That is, we regress Δ Y_i,t on X_i separately in the treated sample and in the untreated sample, restricted to the observations with positive outcomes in period t and t − 1. Since we condition the sample on observations with a positive outcome in period t and t − 1, we require panel data.^[13] Using the fitted functions m1^(x) and m0^(x) , we then calculate fitted values m1^(Xi) and m0^(Xi) . The intensive margin average treatment effect on the treated is then estimated as

where N_T is the number of treated observations with positive outcome in period t and t − 1.^[14]

To conduct inference, we employ a nonparametric quantile bootstrap [15]. From the sample of observations with positive outcomes in period t and t − 1, we repeatedly draw a bootstrap sample of the same sample size. In the bootstrap sample, we estimate the IMATT as described above. This gives a distribution of bootstrap estimated IMATTs: IMATT^t1,…,IMATT^tB , where B is the number of bootstrap replications. We then construct a bootstrap estimated confidence interval as

where q1-α/2* is the (1 − α/2)-percentile of the distribution of bootstrap estimated IMATTs.

5 Empirical Application: Causal Effect of Reaching the Full Retirement Age on Working Hours

We apply the difference-in-difference methodology to estimate the causal intensive margin effect of reaching the full retirement age on working hours of women. We exploit a pension reform in Switzerland taking place in 2004. In this pension reform, the full retirement age (FRA) of women was increased from age 63 to age 64.^[15] This implies that women with year of birth 1941 or earlier reach FRA at age 63, while women with year of birth 1942 or later reach FRA at age 64. We use data from the Swiss Labor Force Survey (SLFS) from 2002–2009. The outcome of interest is working hours, denoted by Y_i,t.^[16] We restrict the sample to women aged 63. Therefore, treatment D_i = 1 for women who have reached FRA (year of birth 1941 or earlier), and D_i = 0 for women who have not reached FRA (year of birth 1942 or later). Since the reform affects individuals only based on their year of birth, assignment to treatment can be assumed to be almost random. We only include a categorical education variable and a dummy for being a Swiss citizen. We consider two estimation samples. The first sample consists of women with year of birth 1941 or 1942. That is, women exactly at the threshold of the pension reform. This sample is cleaner in terms of identification, but the number of observations decreases the power. For this reason we consider a second estimation sample, which includes women with year of birth 1939 to 1946. This sample includes more observations, but might pose a threat to identification if there is a time trend in working hours.

5.1 Discussion: Assumptions

With the exception of time monotonicity at the extensive margin and common support, we cannot directly test the identifying assumptions. Instead, we propose alternative tests that can be used to motivate the identifying assumptions and discuss whether the assumptions are likely to be fulfilled in the context of our empirical application.^[17]

SUTVA: This assumption cannot be tested. There is evidence for spillover effects within couples [16, 17, 18]. That is, the labor supply of one individual depends on whether the spouse has reached FRA. We are aware that this might pose a threat to identification, but assume that the spillover effects are negligible.

No pre-treatment effect: This assumption rules out that people adjust their working hours in anticipation of reaching FRA in the next period. We cannot directly test this assumption. We motivate the assumption by comparing the mean working hours in period t − 1, conditional on having positive working hours in period t and t − 1. The mean in the control group is 23.8 hours, in the treatment group 24.6 hours. A simple Welch two sample t-test does not reject the null hypothesis of equal means (p-value: 0.54). This indicates that the assumption is fulfilled. Moreover, if there is a pre-treatment effect, this effect will likely have the same sign as the treatment effect. As a result, the estimated treatment effect could be interpreted as a lower bound.

Common trend in positive outcomes: This assumption requires that the treatment group would experience the same time trend in working hours in case of no treatment as the control group. We cannot directly test this assumption, but we motivate the assumption by examining the pre-treatment trends of the control and treatment group. In Figure 1, we plot the mean working hours of women with positive hours in period t, t − 1 and t − 2. We observe that the trends between period t − 2 and t − 1 are roughly parallel, indicating that the assumption is fulfilled.

Figure 1

Assessment of Common Trend in Positive Outcomes

Note: Dots indicate the mean working hours of women aged 63 with positive working hours in period t, t-1 and t-2. Bars indicate the 95% normal approximation confidence interval for the mean. Year of birth between 1939 and 1946. Treated is the group which reaches FRA in period t (women with year of birth 1941 or earlier), Control is the group which does not reach FRA in period t (women with year of birth 1942 or later).

No effect of treatment on covariates: In the empirical application, we include a categorical education variable and a dummy for being a Swiss citizen. It seems unlikely that reaching FRA has an effect on these variables. If so, the effect is likely negligible.

Common support: This assumption can be tested. In each covariate cell, we calculate the fraction of treated observations. The results are presented in Table B1 in Appendix B. We observe that there is no covariate cell with only treated observations. Therefore, the assumption is fulfilled.

Treatment monotonicity at the extensive margin: This assumption rules out that people start to work because they reach FRA. There are indeed incentives to take up a job after reaching FRA. For example, part of the earnings are exempted from social security contributions. This increases the net wage. On the other hand, it seems plausible that reaching FRA either has no effect or drives people out of the labor market.

Time monotonicity at the extensive margin: This assumption can be tested. In the treated and control subsamples, we calculate the fraction of individuals with positive working hours in period t, conditional on not working in period t − 1. In the sample of women with year of birth 1939–46, 5% of the treated and 7.4% of the control sample state that they returned to work after having not worked in the period before. This poses a threat to our identification. However, the overall pattern in the age range 60–70 is that people rather leave the labor force as they become older.

6 Conclusion

This paper extends the literature on the identification and estimation of causal intensive margin effects. The intensive margin effect is of interest when subeffects are masked by the total effect. This is the case, for example, when the extensive and intensive margin effect have different signs. We use difference-in-difference methods to identify the causal intensive margin effect. We derive sufficient conditions under which the difference-in-difference estimator on positive outcomes identifies the causal intensive margin effect. We demonstrate that the difference-in-difference estimator on positive outcomes, compared to the standard difference-in-difference estimator, additionally requires time and treatment monotonicity at the extensive margin. We apply the methodology to estimate the causal intensive margin effect of reaching the full retirement age on working hours.